PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE
Nouvelle série, tome 97(111) (2015), 225–231
DOI: 10.2298/PIM140222001N
DOMINATION NUMBER
IN THE ANNIHILATING-IDEAL GRAPHS
OF COMMUTATIVE RINGS
Reza Nikandish, Hamid Reza Maimani, and Sima Kiani
Abstract. Let R be a commutative ring with identity and A(R) be the set of
ideals with nonzero annihilator. The annihilating-ideal graph of R is defined
as the graph AG(R) with the vertex set A(R)∗ = A(R) r {0} and two distinct
vertices I and J are adjacent if and only if IJ = 0. In this paper, we study the
domination number of AG(R) and some connections between the domination
numbers of annihilating-ideal graphs and zero-divisor graphs are given.
1. Introduction
The study of algebraic structures, using the properties of graphs, have become
an exciting research topic in the past twenty years, leading to many interesting
results and questions. There are many papers on assigning a graph to a ring, for
instance see [1, 3, 9, 11].
Throughout this paper, all rings are assumed to be commutative rings with
identity. By Min(R), Z(R) and Nil(R) we denote the set of all minimal prime
ideals of R, the set of all zero-divisors of R and the set of all nilpotent elements of
R, respectively. The socle of ring R, denoted by Soc(R), is the sum of all minimal
ideals of R. If there are no minimal ideals, this sum is defined to be zero. A prime
ideal p is said to be an associated prime ideal of a commutative Noetherian ring
R, if there exists a nonzero element x in R such that p = ann(x). By Ass(R) we
denote the set of all associated prime ideals of R. A T
ring R is said to be reduced, if
it has no nonzero nilpotent element or equivalently P ∈Min(R) P = 0.
For every graph G, we denote by V (G), the vertex set of G. A bipartite graph
is a graph all of whose vertices can be partitioned into two parts U and V such that
every edge joins a vertex in U to one in V . A complete bipartite graph is a bipartite
graph in which every vertex of one part is joined to every vertex of the other part.
If one of the parts is a singleton, then the graph is said to be a star graph. A subset
2010 Mathematics Subject Classification: Primary 13A15; Secondary 05C75.
Key words and phrases: annihilating-ideal graph, zero-divisor graph, domination number,
minimal prime ideal.
The research of H. R. Maimani was partially supported by the grant 91050214 from IPM.
Communicated by Žarko Mijajlović.
225
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NIKANDISH, MAIMANI, AND KIANI
D of V (G) is called a dominating set if every vertex of G is either in D or adjacent
to at least one vertex in D. The domination number of G, denoted by γ(G), is
the number of vertices in a smallest dominating set of G. A total dominating set
of a graph G is a set S of vertices of G such that every vertex is adjacent to a
vertex in S. The total domination number of G, denoted by γt (G), is the minimum
cardinality of a total dominating set. A dominating set of cardinality γ(G) (γt (G))
is called a γ-set (γt -set).
Let R be a ring. The zero-divisor graph of R, Γ(R), is a graph with the vertex
set Z(R) r {0} and two distinct vertices x and y are adjacent if and only if xy = 0.
The concept of the zero-divisor graph was first introduced by Beck (see [5]). We
call an ideal I of R, an annihilating-ideal if there exists a nonzero ideal J of R such
that IJ = 0. We use the notation A(R) for the set of all annihilating-ideals of R.
By the annihilating-ideal graph of R, AG(R), we mean the graph with the vertex
set A(R)∗ = A(R) r {0} such that two distinct vertices I and J are adjacent if and
only if IJ = 0. The annihilating-ideal graph was first introduced in [6] and some
interesting properties of this graph have been studied. In this article, we study
the domination number of the annihilating-ideal graphs. For reduced rings with
finitely many minimal primes and Artinian rings, the domination number of the
annihilating-ideal graphs is given. Also, some relations between the domination
numbers of annihilating-ideal graphs and zero-divisor graphs are studied.
2. Main results
We start with the following remark which completely characterizes all rings for
which either γ(AG(R)) = 1 or γ(Γ(R)) = 1.
Remark 2.1. Let R be a ring. By [6, Theorem 2.2], there is a vertex of AG(R)
which is adjacent to every other vertex if and only if either R ∼
= F × D, where F
is a field and D is an integral domain, or Z(R) is an annihilator ideal. Also, by
[3, Theorem 2.5], there is a vertex of Γ(R) which is adjacent to every other vertex
if and only if either R ∼
= Z2 × D, where D is an integral domain, or Z(R) is an
annihilator ideal. Now, let R be a reduced ring. Then γ(AG(R)) = 1 if and only if
R∼
= F × D, where F is a field and D is an integral domain. Moreover, γ(Γ(R)) = 1
if and only if R ∼
= Z2 × D, where D is an integral domain.
Now, we can state the following proposition.
Proposition 2.1. Let R be a ring. Then we have the following:
(i) If γ(AG(R)) = 1, then γ(Γ(R)) ∈ {1, 2}.
(ii) If γ(Γ(R)) = 1, then γ(AG(R)) = 1.
Proof. The result follows from Remark 2.1 and [10, Proposition 8].
The following result describes the relation between γt (AG(R)) (γt (Γ(R))) and
γ(AG(R)) (γ(Γ(R))).
Theorem 2.1. Let R be a ring. Then
(i) γt (AG(R)) = γ(AG(R)) or γt (AG(R)) = γ(AG(R)) + 1.
DOMINATION NUMBER IN THE ANNIHILATING-IDEAL GRAPHS
227
(ii) γt (Γ(R)) = γ(Γ(R)) or γt (Γ(R)) = γ(Γ(R)) + 1.
Proof. (i) Let γt (AG(R)) 6= γ(AG(R)) and D be a γ-set of AG(R). If
γ(AG(R)) = 1, then
it is clear that γt (AG(R))
Qn = 2. So let γ(AG(R)) > 1
and put k = Max n | ∃ I1 , . . . , In ∈ D s.t. i=1 Ii 6= 0 . Since γt (AG(R)) 6=
Q
γ(AG(R)), we have k > 2. Let I1 , . . . , Ik ∈ D be such that ki=1 Ii 6= 0. Then
Qk
S = { i=1 Ii , ann I1 , . . . , ann Ik } ∪ D r {I1 , . . . , Ik } is a γt -set. Hence γt (AG(R)) =
γ(AG(R)) + 1.
(ii) It is clear by the same argument of part (i).
In the following result we find the total domination number of AG(R).
Theorem 2.2. Let M be the set of all maximal elements of the set A(R). If
|M | > 1, then γt (AG(R)) = |M |.
Proof. Let M be the set of all maximal elements of the set A(R), I ∈ M and
|M | > 1. First we show that I = ann(ann I) and there exists x ∈ R such that
I = ann(x). Let I ∈ M . Then ann I 6= 0 and so there exists 0 6= x ∈ ann I. Hence
I ⊆ ann(ann I) ⊆ ann(x). Thus by the maximality of I, we have I = ann(ann I) =
ann(x). By Zorn’s Lemma it is clear that if A(R) 6= ∅, then M 6= ∅. For any I ∈ M
choose xI ∈ R such that I = ann(xI ). We assert that D = {RxI | I ∈ M } is a total
dominating set of AG(R). Since for every J ∈ A(R) there exists I ∈ M such that
J ⊆ I = ann(xI ), J and RxI are adjacent. Also for each pair I, I ′ ∈ M , we have
RxI RxI ′ = 0. Namely, if there exists x ∈ RxI RxI ′ r {0}, then I = I ′ = ann(x).
Thus γt (AG(R)) 6 |M |. To complete the proof, we show that each element of an
arbitrary γt -set of AG(R) is adjacent to exactly one element of M . Assume to the
contrary, that a vertex K of a γt -set of AG(R) is adjacent to I and I ′ , for I, I ′ ∈ M .
Thus I = I ′ = ann K, which is impossible. Therefore γt (AG(R)) = |M |.
Theorem 2.3. Let R be a ring. Then γt (Γ(R)) 6 γt (AG(R)).
Proof. Let γt (AG(R)) = k. In the light of the proof of Theorem 2.2, there
exists xi ∈ R such that ann(xi ) is a maximal ideal of A(R), for i = 1, . . . , k,
and D = {Rx1 , . . . , Rxk } is a minimum total dominating set of AG(R). It is
easy to check that D′ = {x1 , . . . , xk } is a total dominating set of Γ(R). Thus
γt (Γ(R)) 6 γt (AG(R)).
It is interesting to find some rings for which γt (Γ(R)) = γ(Γ(R)) = γt (AG(R))
= γ(AG(R)). In the next result, we study the domination number of the annihilating-ideal graphs of reduced rings with finitely many minimal primes.
Theorem 2.4. Let R be a reduced ring and | Min(R)| < ∞. If γ(AG(R)) > 1,
then γ(AG(R)) = γt (AG(R)) = | Min(R)|.
Proof. Since R is reduced and γ(AG(R)) > 1, we have | Min(R)| > 1. Suppose that Min(R) = {p1 , . . . , pn }. If n = 2, the result follows from [7, Corollary 2.5]. Therefore, suppose that n > 3. Define b
pi = p1 . . . pi−1 pi+1 . . . pn , for
every i = 1, . . . , n. Clearly, b
pi 6= 0, for every i = 1, . . . , n. Since R is reduced, we
deduce that b
pi pi = 0. Therefore, every pi is a vertex of AG(R). If I is a vertex of
228
NIKANDISH, MAIMANI, AND KIANI
Sn
AG(R), then by [8, Corollary 2.4], I ⊆ Z(R) = i=1 pi . It follows from the Prime
Avoidance Theorem (see [12, Theorem 3.61]) that I ⊆ pi , for some i, 1 6 i 6 n.
Thus pi is a maximal element of A(R), for every i = 1, . . . , n. From Theorem 2.2,
γt (AG(R)) = | Min(R)|. Now, we show that γ(AG(R)) = n. Assume to the contrary, that B = {J1 , . . . , Jn−1 } is a dominating set for AG(R). Since n > 3, the
ideals pi and pj , for i 6= j are not adjacent (from pi pj = 0 ⊆ pk it would follow
that pi ⊆ pk , or pj ⊆ pk which is not true). Because of that, we may assume that
for some k < n − 1, Ji = pi for i = 1, k, but none of the other of ideals from B are
equal to some ps (if B = {p1 , . . . , pn−1 } then pn would be adjacent to some pi , for
i 6= n). So, every ideal in {pk+1 , . . . , pn } is adjacent to an ideal in {Jk+1 , . . . , Jn−1 }.
It follows that for some s 6= t there is an l such that ps Jl = 0 = pt Jl . Since ps * pt ,
it follows that Jl ⊂ pt , so Jl2 = 0, which is impossible, since the ring R is reduced.
So γ(AG(R)) = γt (AG(R)) = | Min(R)|.
Theorem 2.4 leads to the following corollary.
Corollary 2.1. Let R be a reduced ring. If γ(AG(R)) > 1, then the following
are equivalent:
(i) γ(AG(R)) = 2.
(ii) AG(R) is a bipartite graph with two nonempty parts.
(iii) AG(R) is a complete bipartite graph with two nonempty parts.
(iv) R has exactly two minimal primes.
Proof. The result follows from Theorem 2.4 and [7, Corollary 2.5].
In Theorem 15 of [10], it is proved that if R is a finite reduced ring such that
γ(Γ(R)) 6= 1, then γ(Γ(R)) = | Min(R)|. In the next theorem, we prove this result,
where R is not necessarily finite.
Theorem 2.5. Let R be a reduced ring and | Min(R)| < ∞. If γ(Γ(R)) > 1,
then γ(Γ(R)) = γt (Γ(R)) = | Min(R)|.
Proof. Using the notations in the proof of Theorem 2.4, set A = {b
xi | 1 6
i 6 n}, where, for every i, x
bi is an element of b
pi . We show that A is a dominating
set in Γ(R). Since R is reduced, it is easily seen that x
bi is a vertex of Γ(R), for
i = 1, . . . , n. Assume that x ∈
/ A is a vertex of Γ(R). Then x ∈ pi , for some i. The
equality x
bi pi = 0 implies that xb
xi = 0. In the sequel, we prove that γ(Γ(R)) = n.
If n = 2, then [2, Theorem 2.4] completes the proof. Thus assume that n > 3.
Assume to the contrary, the set B = {y1 , . . . , yn−1 } is a dominating
set for Γ(R).
Sn
By the Prime Avoidance Theorem, there exists xi ∈ pi r j=1,j6=i pj . Thus there
exists k, 1 6 k 6 n − 1, such that yk xi = yk xj = 0, for some different i, j,
1 6 i, j 6 n. Since xi ∈
/ pj and xj ∈
/ pi , we have yk ∈ pi ∩ pj . As R is a reduced
ring, we conclude that yk ∈
/ pl , for some l, 1 6 l 6 n. Now, yk xi = 0 ∈ pl implies
that either yk ∈ pl or xi ∈ pl , a contradiction. Clearly, x
bi x
bj = 0, where i 6= j.
Therefore, γ(Γ(R)) = γt (Γ(R)) = | Min(R)|.
Corollary 2.2. Let R be a reduced ring and | Min(R)| < ∞. If γ(AG(R)) 6= 1,
then γ(AG(R)) = γ(Γ(R)).
DOMINATION NUMBER IN THE ANNIHILATING-IDEAL GRAPHS
229
Proof. The result follows from Part (2) of Proposition 2.1 and Theorems 2.4
and 2.5.
In the following theorem the domination number of bipartite annihilating-ideal
graphs is given.
Theorem 2.6. If AG(R) is a bipartite graph, then γ(AG(R)) 6 2.
Proof. If AG(R) is bipartite, it follows from [1, Theorem 27] that one of the
following cases occurs: (a): AG(R) is a star graph; (b): AG(R) is the path of
order 4; (c): Nil(R) = Soc(R). If (a) or (b) happen, then we are done. Suppose
that Nil(R) = Soc(R). If R is reduced, then the result follows from Corollary 2.1.
If R is nonreduced, then [1, Theorem 20] completes the proof.
The next theorem is on the domination number of the annihilating-ideal graphs
of Artinian rings.
Theorem 2.7. Let R be an Artinian ring and R ≇ F1 × F2 , where F1 and F2
are two fields. Then γ(AG(R)) = γt (AG(R)) = | Min(R)|.
Proof. Since R is Artinian, we deduce that each ideal of R is an annihilatingideal. So, the set of maximal elements of A(R) and Max(R) are equal. By [4,
Theorem 8.7], R ∼
= R1 × · · · × Rk , where (Ri , mi ) is an Artinian local ring, for
i = 1, . . . , k. Let Max(R) = {ni = R1 × · · · × Ri−1 × mi × Ri+1 × · · · × Rk | 1 6
i 6 k}. By Theorem 2.2, γt (AG(R)) = | Max(R)|. In the sequel, we prove that
γ(AG(R)) = k. Assume to the contrary, the set {J1 , . . . , Jk−1 } is a dominating
set for AG(R). Since R ≇ F1 × F2 , where F1 and F2 are two fields, we find that
Ji ns = Ji nt = 0, for some i, t, s, where 1 6 i 6 k − 1 and 1 6 t, s 6 k. This means
that Ji = 0, a contradiction.
The condition of R to be an Artinian ring in the previous theorem is necessary;
see the next example.
k[x,y,z]
, where k is a field and x, y and z are indeExample 2.1. Let R = (xy,xz,yz)
terminates. Then γ(AG(R)) = 2 but | Min(R)| > 2.
Theorem 2.7 gives the following immediate corollary.
Corollary 2.3. Let R be an Artinian ring and R ≇ F1 × F2 , where F1 and
F2 are two fields. Then γ(AG(R)) = γ(Γ(R)) = | Min(R)|.
Proof. The result follows from Theorem 2.7 and [10, Theorem 11].
pn1 1 pn2 2
. . . pnmm ,
Example 2.2. Let n be a natural number and n =
where pi ’s
are distinct primes and ni ’s are natural numbers. Then one of the following holds:
(i) γ(AG(Zn )) = 1 if and only if either n = p1 p2 or n = pn1 1 , where n1 > 1.
(ii) γ(AG(Zn )) = 2 if and only if m = 2 and either n1 > 1 or n2 > 1.
(iii) If m > 3, then γ(AG(Zn )) = m.
The following theorem provides an upper bound for the domination number of
the annihilating-ideal graph of a Noetherian ring.
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NIKANDISH, MAIMANI, AND KIANI
Theorem 2.8. If R is a Notherian ring, then γ(AG(R)) 6 | Ass(R)| < ∞.
Proof. By [12, Remark 9.33], | Ass(R)| < ∞. Let Ass(R) = {p1 , . . . , pn } and
pi = ann(xi ), for some xi ∈ R and for every i = 1, . . . , n. Set A = {Rxi | 1 6 i 6 n}.
We show that A is a dominating set of AG(R). Clearly, every Rxi is a vertex of
AG(R), for i = 1, . . S
. , n. If I is a vertex of AG(R), then [12, Corollary 9.36] implies
that I ⊆ Z(R) = ni=1 pi . It follows from the Prime Avoidance Theorem that
I ⊆ pi , for some i, 1 6 i 6 n. Thus IRxi = 0, as desired.
The inequality in the above theorem may be strict; see the next example.
Example 2.3. Let R = (xk[x,y]
2 ,xy) , where k is a field and x, y are indeterminate.
Then Ass(R) = {(x), (x, y)} but γ(AG(R)) = 1 (Indeed the vertex (x) is adjacent
to all other vertices).
Remark 2.2. If R is Noetherian, then by a similar argument to that of the
proof of Theorem 2.8, one can show that γ(Γ(R)) 6 | Ass(R)|.
We end the paper with the following result about the domination number of
the annihilating-ideal graph of a finite direct product of rings.
Theorem 2.9. For a ring R, which is a product of two (nonzero) rings, one
of the following holds:
(i) If R ∼
= F × D, where F is a field and D is an integral domain, then
γ(AG(R)) = 1.
(ii) If R ∼
= D1 × D2 , where D1 and D2 are integral domains which are not
fields, then γ(AG(R)) = 2.
(iii) If R ∼
= R1 × D, where R1 is a ring which is not integral domain and D is
an integral domain, then γ(AG(R)) = γ(AG(R1 )) + 1.
(iv) If R ∼
= R1 × R2 , where R1 and R2 are two rings which are not integral
domains, then γ(AG(R)) = γ(AG(R1 )) + γ(AG(R2 )).
Proof. Parts (i) and (ii) are clear.
(iii) With no loss of generality, one can assume that γ(AG(R1 )) < ∞. Suppose
that γ(AG(R1 )) = n and {I1 , . . . , In } is a minimal dominating set of AG(R1 ). It
is not hard to see that {I1 × 0, . . . , In × 0, 0 × D} is the smallest dominating set of
AG(R).
(iv) We may assume that γ(AG(R1 )) = m and γ(AG(R2 )) = n, for some
positive integers m and n. Let {I1 , . . . , Im } and {J1 , . . . , Jn } be two minimal dominating sets in AG(R1 ) and AG(R1 ), respectively. It is easily seen that {I1 × 0,
. . . , Im × 0, 0 × J1 . . . 0 × Jn } is the smallest dominating set in AG(R).
Acknowledgements. The authors are deeply grateful to the referee for his/her
valuable suggestions.
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Department of Mathematics
Jundi-Shapur University of Technology
Dezful, Iran
r.nikandish@jsu.ac.ir
Mathematics Section
Department of Basic Sciences
Shahid Rajaee Teacher Training University
Tehran, Iran
and
School of Mathematics
Institute for Research in Fundamental Sciences (IPM)
Tehran, Iran
maimani@ipm.ir
Department of Mathematics
Science and Research Branch
Islamic Azad University
Tehran, Iran
s.kiani@srbiau.ac.ir
(Received 01 05 2013)
(Revised 01 01 2014)